Surface Area and Volume Measurements
3D print toolbox

In the 3D print toolbox add-on (it's already bundled with the official verion of Blender, you just have to enable it) there are two buttons (you can fin the right on the top of the tab), one for the volume and one for the surface (area)

Results are shown in the latest line of the panel once one of the two buttons is pressed.



The commands take into account the whole object, so if you need to get the surface of one part of your object only, you'll have to separate the faces into another object.